%I #60 Jan 05 2021 05:03:14
%S 1,-2,1,-2,4,-4,5,-6,9,-12,13,-16,21,-26,29,-36,46,-54,62,-74,90,-106,
%T 122,-142,171,-200,227,-264,311,-358,408,-470,545,-626,709,-810,933,
%U -1062,1198,-1362,1555,-1760,1980,-2238,2536,-2858,3205,-3602,4063,-4560,5092,-5704,6400,-7150,7966
%N Expansion of Product_{m >= 1} (1 + q^m)^(-2).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C McKay-Thompson series of class 24J for the Monster group.
%D T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.
%H Seiichi Manyama, <a href="/A022597/b022597.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H D. Foata and G.-N. Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub81.html">Jacobi and Watson Identities Combinatorially Revisited</a>
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 13.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of q^(1/12) * (eta(q) / eta(q^2))^2 in powers of q.
%F Euler transform of period 2 sequence [ -2, 0, ...]. - _Michael Somos_, Sep 10 2005
%F Expansion of chi(-x)^2 in powers of x where chi() is a Ramanujan theta function.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A022567.
%F G.f.: Product_{k>0} (1 + x^k)^-2.
%F Convolution square of A081362. Convolution inverse of A022567.
%F a(n) = (-1)^n * A073252(n).
%F a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017
%F G.f.: exp(-2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%e G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^4 - 4*x^5 + 5*x^6 - 6*x^7 + 9*x^8 + ...
%e T24J = 1/q - 2*q^11 + q^23 - 2*q^35 + 4*q^47 - 4*q^59 + 5*q^71 - 6*q^83 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2, {x, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}]^-2, {x, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^2, n))}; /* _Michael Somos_, Sep 10 2005 */
%Y Cf. A022567, A073252, A081362.
%Y Cf. A089814 (expansion of Product_{k>=1}(1-q^(10k-5))^2).
%Y Column k=2 of A286352.
%K sign,nice,easy
%O 0,2
%A _N. J. A. Sloane_